Covering Planar Lattices with Interior-Disjoint Unit Disks

2026-07-22

AI Summary

This study investigates the feasibility of covering periodic point sets in the plane—specifically triangular, square, and honeycomb lattices—with non-overlapping unit disks, determining the ranges of inter-point distances that permit complete coverage. By constructing single- and multi-family periodic covering patterns centered at lattice vertices, face centers, and off-lattice points, and combining symmetry analysis with computer-assisted verification, the work systematically characterizes the conditions for full coverage. The main contributions include identifying several new intervals of coverable spacing for the triangular lattice, establishing the first effective covering scheme for honeycomb point sets, and correcting an overlap error in prior results for the square lattice—thereby recovering known coverable intervals and discovering additional ones.

Abstract

We study an infinite variant of the coin-covering problem for periodic point sets in the plane. Given a point set of spacing $d$, we ask whether all of its points can be covered by pairwise non-overlapping unit disks. We consider the triangular lattice, the square lattice, and the honeycomb point set, and construct periodic motif patterns that certify several intervals of coverable spacings. For the triangular lattice, our constructions include single-family patterns with vertex, face, and off-lattice realizing centers, as well as multi-family patterns. For the honeycomb point set, additional native motifs fill gaps left by the triangular-lattice constructions. For the square lattice, we revisit the constructions of Alm et al., identify an unintended overlap in one motif realization, and give new patterns that recover part of the affected interval and establish an additional coverability interval.

Problem

• coin-covering problem
• periodic point sets
• unit disks
• planar lattices
• non-overlapping covering

Innovation

• periodic motif patterns
• interior-disjoint unit disks
• lattice covering
• coin-covering problem
• coverability intervals